It was proven in [1]} that the Haussdorff dimension of the cut locus on a closed Riemannian manifold \((N,g)\)  is locally an integer that does not exceed \(\dim N -1\) . Moreover \(T(p)\)  is a smooth hyper surface of \(N\)  and the Hausdorff dimension of \(Q(p)\cup L(p)\)  does not exceed \(\dim N-2\) . In this paper we will extended these results for manifolds with strictly convex boundary. The main result of this section is as follows:
